Well, we can assume that $V\simeq F^n$, and so the affine lines parallel to the basis vectors are given by the system of $n$ equations $x_1 = c_1,\ldots, x_i = x_i, \ldots, x_n = c_n$, for some $i\in \{1,\ldots,n\}$. What about some sort of ruled surface?
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David RobertsJan 4 '12 at 23:48

Also, you can take $S \subset \mathbb{P}^{n-1}$, and translate the problem to projective space.
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David RobertsJan 4 '12 at 23:49

In 2), should "For every affine" be replaced with "Every affine"? $\;$
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Ricky DemerJan 5 '12 at 1:47

Start with the plane with normal direction $(1,1,1)$. Tilt each radial ray of that plane by an angle $\alpha = \epsilon \sin 3\phi$ towards $(1,1,1)$ (or away if negative), where $\phi$ is the angle the ray makes with the ray through, say, $(1,-1,0)$. For small $\epsilon$, the resulting surface is still a graph of a single valued function $x=f_1(y,z)$ or $y=f_2(z,x)$ or $z=f_3(x,y)$.